4. Suppose that lim s, =0 and suppose that (t, ) is a bounded sequence. Show that lim s,t, = 0 %3D
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
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Math 4303 Homework Section 4.1 Convergence
1. Write out the first seven terms of the sequence dn
2n +1
%3D
3n-1
2. Using only Definition 4.1.2 (N-ɛ definition of convergence), prove the following:
k
a] lim
- = 0
kER
4n +1
b] lim
no n+3
= 4
6n? +3n
= 3
no 2n? -5
c] lim
sin( n)
=D0
d] lim
3. Prove or provide a counterexample for the following:
a] If (s,) converges then (s) converges.
b] If (s,) converges then (s,) converges.
c] lim s, = 0 iff lim |s„ =0
4. Suppose that lim s, =0 and suppose that (t, ) is a bounded sequence. Show that lim s,t, = 0
5. Suppose that (x,), (y, ), and (=,) are sequences such that x, Sy, S=, n and lim x, = L = lim =
Show that lim y, = L
6. Suppose that lim s. =s where s > 0. Prove that there exists a natural number N, such that
if n> N, then s, > 0
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